Two-Dimensional Asymmetric Multiferroics: Unique Way toward Strong Magnetoelectric Coupling and Multistate Memory

Two-dimensional (2D) materials have provided a fascinating platform for exploring novel multiferroics and emergent magnetoelectric coupling mechanisms. Here, a novel 2D asymmetric multiferroic based on Janus 2D multiferroic MXene-analogous oxynitrides (InTlNO2) is presented by using first-principles calculations. We find three inequivalent phases for InTlNO2, including two metallic phases (p1 and p2) and one semiconducting phase (p3) with a band gap of 0.88 eV. All phases are room-temperature multiferroics with different Curie temperatures, leading to tunability by phase transitions. We show that there is a 90° rotation of the magnetic anisotropy easy axis between p1 and p2, where p1 favors the in-plane and p2 the out-of-plane easy axis. Therefore, the magnetic anisotropy can be tuned by reversing the out-of-plane polarization. Our strategy provides a unique way toward strong magnetoelectric coupling and multistate memory.

Table S1.In-plane elastic constants of phase1, phase2 and phase 3 (kBar).Considering the nearest-neighbor, second-nearest-neighbor and third-nearest-neighbor magnetic exchange interaction, the spin Hamiltonian can be defined as [1][2][3] :     is the  single-site magnetic anisotropy energy parameter.The total energies of these four spin configurations can be described as: Where represents the energy without magnetic coupling., , and  0      1 are the energies of FM. sAFM, zAFM1 and zAFM2 spin configurations.Therefore,  2 one can obtain: Assuming that S = 1/2 for all, we can obtain the values of these exchange coupling constants.Due to the asymmetric energy barriers, six kinds of transitions were considered (p1 to p2, p2 to p1, p1 to p3, p3 to p1, p2 to p3 and p3 to p2).According to the CI-NEB curve and the calculated electric polarization, p1 can switch to p2 directly under a large positive electric field, while a medium negative electric field is required to transform p2 back to p1 (Figure S9).For p1 and p3, small external field can achieve the transformation between them.p1 can directly cross p3 and transform into p2 if the electric field is large enough to overcome the energy barrier between p3 and p2 (Figure S10).The scenario for p2 and p3 may be a little more complicated (Figure S11).Relatively large positive electric field is demanded to directly transform p3 to p2, while a negative electric field may always transform p2 to p1 first since the energy barrier for p3 to p1 is much smaller.Therefore, if we want to transform p2 into p3, a negative electric field is required first to switch p2 to p1, and then a smaller positive electric field can transform p1 to p3 (p2→p1→p3).

Figure S6 .
Figure S6.A schematic diagram of magnetic exchange interaction in p3 (D: direct exchange, S: super exchange).

Figure S7 .
Figure S7.The phonon spectra of (a) p12 and (b) p23 (The imaginary modes are marked by red dashed box).

Figure S9 .
Figure S9.(a) p1 to p2 and (b) p2 to p1 (The number of symbols is directly proportional to the electric field strength same as below).

Figure S11 .
Figure S11.(a) p3 to p2 and (b) p2 to p3 (p2→p1→p3).Due to the asymmetric energy barriers, six kinds of transitions were considered (p1 to p2, p2 to p1, p1 to p3, p3 to p1, p2 to p3 and p3 to p2).According to the CI-NEB curve and the calculated electric polarization, p1 can switch to p2 directly under a large positive electric field, while a medium negative electric field is required to transform p2 back to p1 (FigureS9).For p1 and p3, small external field can achieve the transformation between them.p1 can directly cross p3 and transform into p2 if the electric field is large enough to overcome the energy barrier between p3 and p2 (FigureS10).The scenario for p2 and p3 may be a little more complicated (FigureS11).Relatively large positive electric field is demanded to directly transform p3 to p2, while a negative electric field may always transform p2 to p1 first since the energy barrier for p3 to p1 is much smaller.Therefore, if we want to transform p2 into p3, a negative electric field is required first to switch p2 to p1, and then a smaller positive electric field can transform p1 to p3 (p2→p1→p3).

Figure S15 .
Figure S15.Angle-dependent magnetic anisotropic energy of p3 with magnetic moments lying on xy (violet), xz (olive) and yz (orange) planes.The definition of angles is shown in the left.

Figure S18 .
Figure S18.Orbital-resolved ΔE soc (deviation between in-plane direction and easy axis) of d orbitals for p3: (a) Tl and (b) In (Positive values favor the easy axis, while negative values favor in-plane direction).

Figure S19 .
Figure S19.Orbital-resolved ΔE soc (deviation between in-plane direction and easy axis) of p orbitals for p3: (a) Tl and (b) In (Positive values favor the easy axis, while negative values

Table S2 .
Relative energy of different InTlNO 2 (the energy of p1 is set to 0).